Appears in collection : Geometry in non-positive curvature and Kähler groups

The JSJ (for groups) was originally constructed to study the automorphisms and the cyclic splittings of a (torsion-free) hyperbolic group. Such a structure theory was needed to complete the solution of the isomorphism problem for (torsion-free) hyperbolic groups.

Later, the JSJ was generalized to all finitely presented groups. In this generality it encodes the splittings but not all the automorphisms.

We further generalize the JSJ decomposition to study automorphisms of groups that act on products of hyperbolic spaces, and more generally to study automorphisms of (some) hierarchically hyperbolic groups (e.g. right angled Artin groups). The object that we construct can be viewed as a higher rank JSJ decomposition.